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Chapter 11: Problem 44

Use a symbolic integration utility to find the indefinite integral. $$ \int(1+3 t) t^{2} d t $$

Short Answer

Expert verified
The indefinite integral of the function is \( \frac{1}{3}t^3 + \frac{3}{4}t^4 + C \)

Step by step solution

01

Identify the function within the integral

The function within the integral sign in this case is \((1+3t)t^{2}\). There is an implicit multiplication between \(t^{2}\) and the expression in the parentheses \(1+3t\). This means that the function can be expanded.
02

Expand the expression within the integral

By expanding the expression, the function becomes \(t^{2} + 3t^{3}\). The function can now be broken down and each term can be integrated separately.
03

Use the Power Rule for Integration

The power rule for integration states that the integral of \(x^n\), with respect to \(x\), where \(n\) is any real number except -1, is \(\frac{1}{n+1}x^{n+1}\). Apply this rule to each term of the function to find the antiderivative. This gives \( \frac{1}{3}t^3 + \frac{3}{4}t^4 \).

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Most popular questions from this chapter

Find the consumer and producer surpluses. $$ p_{1}(x)=300-x \quad p_{2}(x)=100+x $$

Use the Trapezoidal Rule with \(n=4\) to approximate the definite integral. $$ \int_{0}^{4} \sqrt{1+x^{2}} d x $$

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{-1}^{1}\left[\left(1-x^{2}\right)-\left(x^{2}-1\right)\right] d x $$

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ \begin{aligned} &y=x^{2}-4 x+3, y=3+4 x-x^{2} \\ &y=4-x^{2} \cdot y=x^{2} \end{aligned} $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(x)=\sqrt{x}, \quad[0,1] $$
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